A bank is a financial institution which stores and lends money usually on both an individual and a corporate level.
While banks can charge fees for services, mostly people depositing money in a bank will seem to see the amount of that money slightly increase over a long time due to interest rates that the bank will pay. Popular avenues for storing money are in checking or savings accounts. Generally speaking checking accounts allow the writing of checks while savings accounts will not. Savings accounts will generally pay a greater interest rate.
Banks can also offer certificates of deposit where the money is tied up for a set length of time earning a set interest rate. Generally speaking, the longer the period of time the money is tied up, the greater the interest rate it will earn.
Some banks now also offer full service investing, where they will have a vendor come into the bank and offer access to mutual funds and other investing opportunities that the banks themselves do not directly offer.
In theory, banks should offer interest rates that are higher than the rate of inflation. However, in certain economic conditions, that fails to be true and the customer loses money (in inflation-adjusted terms) for storing money in a bank. Customers will still use banks in those conditions rather than hiding money at home or investing in a hard assets because of the liquidity and security that banks provide.
When banks make loans they charge a rate of interest to do so. That rate is always higher than the rate they pay out for deposits. The most common form of loan is a home loan, although banks can also make personal loans, business loans or provide home equity lines based upon the equity in a home after all loans against the property are subtracted. Loan availability and loan rates usually based upon yearly household income and credit scores.
Although banks are not part of the government (see Federal Reserve Bank), in every major country accounts with a bank are insured by the government up to a certain amount. The goal, as emphasized in the Financial Crisis of 2008, was to prevent "bank runs" in which people overnight lose confidence and demand all their deposits back.